---
_id: '9817'
abstract:
- lang: eng
  text: Elastic bending of initially flat slender elements allows the realization
    and economic fabrication of intriguing curved shapes. In this work, we derive
    an intuitive but rigorous geometric characterization of the design space of plane
    elastic rods with variable stiffness. It enables designers to determine which
    shapes are physically viable with active bending by visual inspection alone. Building
    on these insights, we propose a method for efficiently designing the geometry
    of a flat elastic rod that realizes a target equilibrium curve, which only requires
    solving a linear program. We implement this method in an interactive computational
    design tool that gives feedback about the feasibility of a design, and computes
    the geometry of the structural elements necessary to realize it within an instant.
    The tool also offers an iterative optimization routine that improves the fabricability
    of a model while modifying it as little as possible. In addition, we use our geometric
    characterization to derive an algorithm for analyzing and recovering the stability
    of elastic curves that would otherwise snap out of their unstable equilibrium
    shapes by buckling. We show the efficacy of our approach by designing and manufacturing
    several physical models that are assembled from flat elements.
acknowledgement: "We thank the anonymous reviewers for their generous feedback, and
  Michal Piovarči for his help in producing the supplemental video. This project has
  received funding from the European Research Council (ERC) under the European Union’s
  Horizon 2020 research and innovation programme (grant agreement No 715767).\r\n"
article_number: '126'
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Hafner, Christian
  id: 400429CC-F248-11E8-B48F-1D18A9856A87
  last_name: Hafner
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
citation:
  ama: Hafner C, Bickel B. The design space of plane elastic curves. <i>ACM Transactions
    on Graphics</i>. 2021;40(4). doi:<a href="https://doi.org/10.1145/3450626.3459800">10.1145/3450626.3459800</a>
  apa: 'Hafner, C., &#38; Bickel, B. (2021). The design space of plane elastic curves.
    <i>ACM Transactions on Graphics</i>. Virtual: Association for Computing Machinery.
    <a href="https://doi.org/10.1145/3450626.3459800">https://doi.org/10.1145/3450626.3459800</a>'
  chicago: Hafner, Christian, and Bernd Bickel. “The Design Space of Plane Elastic
    Curves.” <i>ACM Transactions on Graphics</i>. Association for Computing Machinery,
    2021. <a href="https://doi.org/10.1145/3450626.3459800">https://doi.org/10.1145/3450626.3459800</a>.
  ieee: C. Hafner and B. Bickel, “The design space of plane elastic curves,” <i>ACM
    Transactions on Graphics</i>, vol. 40, no. 4. Association for Computing Machinery,
    2021.
  ista: Hafner C, Bickel B. 2021. The design space of plane elastic curves. ACM Transactions
    on Graphics. 40(4), 126.
  mla: Hafner, Christian, and Bernd Bickel. “The Design Space of Plane Elastic Curves.”
    <i>ACM Transactions on Graphics</i>, vol. 40, no. 4, 126, Association for Computing
    Machinery, 2021, doi:<a href="https://doi.org/10.1145/3450626.3459800">10.1145/3450626.3459800</a>.
  short: C. Hafner, B. Bickel, ACM Transactions on Graphics 40 (2021).
conference:
  end_date: 2021-08-13
  location: Virtual
  name: 'SIGGRAF: Special Interest Group on Computer Graphics and Interactive Techniques'
  start_date: 2021-08-09
date_created: 2021-08-08T22:01:26Z
date_published: 2021-07-19T00:00:00Z
date_updated: 2026-04-21T22:30:03Z
day: '19'
ddc:
- '516'
department:
- _id: BeBi
doi: 10.1145/3450626.3459800
ec_funded: 1
external_id:
  isi:
  - '000674930900091'
file:
- access_level: open_access
  checksum: 7e5d08ce46b0451b3102eacd3d00f85f
  content_type: application/pdf
  creator: chafner
  date_created: 2021-10-18T10:42:15Z
  date_updated: 2021-10-18T10:42:15Z
  file_id: '10150'
  file_name: elastic-curves-paper.pdf
  file_size: 17064290
  relation: main_file
  success: 1
- access_level: open_access
  checksum: 0088643478be7c01a703b5b10767348f
  content_type: application/pdf
  creator: chafner
  date_created: 2021-10-18T10:42:22Z
  date_updated: 2021-10-18T10:42:22Z
  file_id: '10151'
  file_name: elastic-curves-supp.pdf
  file_size: 547156
  relation: supplementary_material
file_date_updated: 2021-10-18T10:42:22Z
has_accepted_license: '1'
intvolume: '        40'
isi: 1
issue: '4'
keyword:
- Computing methodologies
- shape modeling
- modeling and simulation
- theory of computation
- computational geometry
- mathematics of computing
- mathematical optimization
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: 24F9549A-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '715767'
  name: 'MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and
    Modeling'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
  link:
  - description: News on IST Website
    relation: press_release
    url: https://ist.ac.at/en/news/designing-with-elastic-structures/
  record:
  - id: '12897'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: The design space of plane elastic curves
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 40
year: '2021'
...